In Exercises 47–54, use the figures to find the exact value of...

Question

In Exercises 47–54, use the figures to find the exact value of each trigonometric function. cos(α/2)

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the trigonometric function. Using the provided figure, we want to find the value of the cosine of data divided by two. We are given a right triangle. Pr Q with a right angle at R and theta at angle Q side pr measures 28 side. QR measures 45. We have four answer choices. A seven radical 53 divided by 53 B seven radical 53 divided by 54 C radical 53 divided by 53 D seven divided by 53. First, I want to write the formula for the cosine of theta divided by two and that is the square root of one plus the cosine of theta divided by two. So using my diagram, I have a lot of information but unfortunately not enough yet to find the cosine of data. So the cosine of theta I know is the adjacent side divided by the hypotenuse. So the site adjacent to theta is 45. So we know that part, we unfortunately don't know our hypotenuse yet. So I'm gonna label that a and then I'm gonna use the Pythagorean theorem. To solve for it. So my legs squared, so 28 squared plus 45 squared, S equal the hypotenuse squared. So A squared, 28 squared is 784 plus 45 square. So 2025 and that equals A squared, adding those together. I get 2809 equaling A squared. I take the square root of both sides and get the A equals 53. So this means my hypotenuse equals 53. And I can find the value for the cosine of theta as 45 divided by 53. So let's plug that into our formula to find the cosine of theta divided by two. So we have the square root of one plus 45 divided by 53 divided by two, combining my fractions. Still under my radical, I will get 98 53rd divided by two. So if I rewrite this, the cosine of theta divided by two equals the square root of 98 divided by 53 multiplied by the reciprocal of two, which is one half. I'm gonna cross reduce the two and the 98. Now I get the square root of 49 divided by 53. Recalling my rules about fractions and radicals. I'm going to rewrite this as the square root of 49 divided by the square root of 53. The square to 49 is seven. The square to 53 is not a pretty number. So we are gonna leave it under the radical as the square to 53. We do however, need to rationalize our denominator. So we'll be multiplying both the numerator and the denominator by the square of 53. And when we do so we get seven radical 53 divided by 53. And that's as simplified as this can get. So this is our solution. That's the exact value. So seven radical 53 then divided by 53 and that's answer choice. A have a nice day.

In Exercises 47–54, use the figures to find the exact value of each trigonometric function. cos(α/2)

Key Concepts

Trigonometric Functions

Unit Circle and Angle Measures

Reference Angles and Quadrants

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